Efficient circular arc interpolation based on active tolerance control

In this paper, we present an efficient sub-optimal algorithm for fitting smooth planar parametric curves by G¹ arc splines. To fit a parametric curve by an arc spline within a prescribed tolerance, we first sample a set of points and tangents on the curve adaptively as well as with enough density, so that an interpolation biarc spline curve can be with any desired high accuracy. Then, we construct new biarc curves interpolating local triarc spirals explicitly based on the control of permitted tolerances. To reduce the segment number of fitting arc spline as much as possible, we replace the corresponding parts of the spline by the new biarc curves and compute active tolerances for new interpolation steps. By applying the local biarc curve interpolation procedure recursively and sequentially, the result circular arcs with no radius extreme are minimax-like approximation to the original curve while the arcs with radius extreme approximate the curve parts with curvature extreme well too, and we obtain a near optimal fitting arc spline in the end. Even more, the fitting arc spline has the same end points and end tangents with the original curve, and the arcs will be jointed smoothly if the original curve is composed of several smooth connected pieces. The algorithm is easy to be implemented and generally applicable to circular arc interpolation problem of all kinds of smooth parametric curves. The method can be used in wide fields such as geometric modeling, tool path generation for NC machining and robot path planning, etc. Several numerical examples are given to show the effectiveness and efficiency of the method.     

Fairing an arc spline and designing with G 2 PH quintic spiral transitions

This paper describes a method to smooth an arc spline. Arc splines are G ¹ continuous segments made of circular arcs and straight lines. We have proposed a smooth version of the arc spline by replacing its parts with C-, S-, and J-shaped spiral transitions, stitched with G ² continuity, by using a single segment of Pythagorean hodograph quintic function. Use of a single polynomial function rather than two has the benefit that designers have fewer entities to deal with. Spiral transitions are important in manufacturing industries because of their use in the cutting paths for numerically controlled cutting machinery, highway or railway designing, non-holonomic robot path planning and spur gear designing.     

Optimal arc spline approximation

We present a method for approximating a point sequence of input points by a G¹$G^1$-continuous (smooth) arc spline with the minimum number of segments while not exceeding a user-specified tolerance. Arc splines are curves composed of circular arcs and line segments (shortly: segments). For controlling the tolerance we follow a geometric approach: We consider a simple closed polygon P and two disjoint edges designated as the start s and the destination d. Then we compute a SMAP (smooth minimum arc path), i.e. a smooth arc spline running from s to d in P with the minimally possible number of segments. In this paper we focus on the mathematical characterization of possible solutions that enables a constructive approach leading to an efficient algorithm.     

G 2 interpolation and blending on surfaces

We introduce a method for curvature-continuous (G ²) interpolation of an arbitrary sequence of points on a surface (implicit or parametric) with prescribed tangent and geodesic curvature at every point. The method can also be used forG ² blending of curves on surfaces. The interpolation/blending curve is the intersection curve of the given surface with a functional spline (implicit) surface. For the construction of blending curves, we derive the necessary formulas for the curvature of the surfaces. The intermediate results areG ² interpolation/blending methods in IR².     

Curve fitting and fairing using conic splines

We present an efficient geometric algorithm for conic spline curve fitting and fairing through conic arc scaling. Given a set of planar points, we first construct a tangent continuous conic spline by interpolating the points with a quadratic Bézier spline curve or fitting the data with a smooth arc spline. The arc spline can be represented as a piecewise quadratic rational Bézier spline curve. For parts of the G¹ conic spline without an inflection, we can obtain a curvature continuous conic spline by adjusting the tangent direction at the joint point and scaling the weights for every two adjacent rational Bézier curves. The unwanted curvature extrema within conic segments or at some joint points can be removed efficiently by scaling the weights of the conic segments or moving the joint points along the normal direction of the curve at the point. In the end, a fair conic spline curve is obtained that is G² continuous at convex or concave parts and G¹ continuous at inflection points. The main advantages of the method lies in two aspects, one advantage is that we can construct a curvature continuous conic spline by a local algorithm, the other one is that the curvature plot of the conic spline can be controlled efficiently. The method can be used in the field where fair shape is desired by interpolating or approximating a given point set. Numerical examples from simulated and real data are presented to show the efficiency of the new method.     

A class of general quartic spline curves with shape parameters

With a support on four consecutive subintervals, a class of general quartic splines are presented for a non-uniform knot vector. The splines have C² continuity at simple knots and include the cubic non-uniform B-spline as a special case. Based on the given splines, piecewise quartic spline curves with three local shape parameters are given. The given spline curves can be C²∩G³ continuous by fixing some values of the curve?s parameters. Without solving a linear system, the spline curves can also be used to interpolate sets of points with C² continuity. The effects of varying the three shape parameters on the shape of the quartic spline curves are determined and illustrated.     

C3连续的单位四元数插值样条曲线

目的 构造一类C³连续的单位四元数插值样条曲线,证明它的插值性和连续性,并把它应用于刚体关键帧动画设计中。方法 利用R³空间中插值样条曲线的5次多项式调配函数的累和形式构造了S³空间中单位四元数插值样条曲线,它不仅能精确通过一系列给定的方向,而且能生成C³连续的朝向曲线。结果 与Nielson的单位四元数均匀B样条插值曲线的迭代构造方法相比,所提方法避免了为获取四元数B样条曲线控制顶点对非线性方程组迭代求解的过程,提高了运算效率;与单位四元数代数三角混合插值样条曲线的构造方法（Su方法）相比,所提方法只用到多项式基,运算速度更快。本例中创建关键帧动画所需的时间与Nielson方法和Su方法相比平均下降了73%和33%。而且,相比前两种方法,所提方法产生的四元数曲线连续性更高,由C²连续提高到C³连续,这意味着动画中刚体的朝向变化更加自然。结论 仿真结果表明,本文方法对刚体关键帧动画设计是有效的,对实时性和流畅性要求高的动画设计场合尤为适用。     

C2 interpolation of spatial data subject to arc-length constraints using Pythagorean–hodograph quintic splines

In order to reconstruct spatial curves from discrete electronic sensor data, two alternative C² Pythagorean–hodograph (PH) quintic spline formulations are proposed, interpolating given spatial data subject to prescribed constraints on the arc length of each spline segment. The first approach is concerned with the interpolation of a sequence of points, while the second addresses the interpolation of derivatives only (without spatial localization). The special structure of PH curves allows the arc-length conditions to be expressed as algebraic constraints on the curve coefficients. The C² PH quintic splines are thus defined through minimization of a quadratic function subject to quadratic constraints, and a close starting approximation to the desired solution is identified in order to facilitate efficient construction by iterative methods. The C² PH spline constructions are illustrated by several computed examples.     

Constructing G 1 continuous curve on a free-form surface with normal projection

In this paper, we develop a new method for G ¹ continuous interpolation of an arbitrary sequence of points on an implicit or parametric surface with a specified tangent direction at every point. Based on the normal projection method, we design a G ¹ continuous curve in three-dimensional space and then project orthogonally the curves onto the given surface. With the techniques in classical differential geometry, we derive a system of differential equations characterizing the projection curve. The resulting interpolation curve is obtained by numerically solving the initial-value problems for a system of first-order ordinary differential equations in the parametric domain associated to the surface representation for a parametric case or in three-dimensional space for an implicit case. Several shape parameters are introduced into the resulting curve, which can be used in subsequent interactive modification such that the shape of the resulting curve meets our demand. The presented method is independent of the geometry and parameterization of the base surface, and numerical experiments demonstrate that it is effective and potentially useful in surface trim, robot, patterns design on surface and other industrial and research fields.     

Function Representation of Solids Reconstructed from Scattered Surface Points and Contours

This paper presents a novel approach to the reconstruction of geometric models and surfaces from given sets of points using volume splines. It results in the representation of a solid by the inequality f(x,y,z) ≥ 0. The volume spline is based on use of the Green's function for interpolation of scalar function values of a chosen “carrier” solid. Our algorithm is capable of generating highly concave and branching objects automatically. The particular case where the surface is reconstructed from cross-sections is discussed too. Potential applications of this algorithm are in tomography, image processing, animation and CAD for bodies with complex surfaces.     

参数曲面上的插值与混合

如何表示曲面上的曲线,在处理诸如数控加工中的路径设计以及CAD/CAM等领域频繁出现的曲面裁剪问题时显得日益重要.给出了数据点的切方向(切方向及曲率向量或测地曲率值)指定而G¹连续(G²连续)插值曲面上任意点列的方法.作为曲面上曲线插值问题的特例,还讨论了曲面上曲线的混合问题.基本思想是借助于微分几何的有关结论,曲面上曲线的插值问题被转化为其参数平面上类似的曲线插值问题.该方法能够用二维隐式方程来表示曲面上的插值曲线,从而把在显示该曲线时所面对的曲面求交的几何问题转化为计算隐式曲线的代数问题.实验证明该方法是可行的,而且适用于CAD/CAM及计算机图形学等领域.     

单圆弧样条保形插值算法

该文以插值具有偶数个点的闭多边形为例提出了一种新的圆弧样条插值算法。这种算法具有以下3个特点：（1）生成的圆弧样条曲线具有保形的特点；（2）圆弧样条中圆弧的段数与型值点个数相同。（3）圆弧段之间的连接点不一定在插值的型值点上，这样就能用更多的自由度来控制拟合曲线的形状。同此文中还提出了一个优化的算法来得到光顺的插值曲线，同时还给出了几个例子加以说明。     

C<Superscript>2</Superscript> quartic spline surface interpolation

This paper discusses the problem of constructing C² quartic spline surface interpolation. Decreasing the continuity of the quartic spline to C² offers additional freedom degrees that can be used to adjust the precision and the shape of the interpolation surface. An approach to determining the freedom degrees is given, the continuity equations for constructing C² quartic spline curve are discussed, and a new method for constructing C² quartic spline surface is presented. The advantages of the new method are that the equations that the surface has to satisfy are strictly row diagonally dominant, and the discontinuous points of the surface are at the given data points. The constructed surface has the precision of quartic polynomial. The comparison of the interpolation precision of the new method with cubic and quartic spline methods is included.     

A constrained guided G continuous spline curve

A method for drawing a guided G¹ continuous planar spline curve that falls within a closed boundary is presented. The curve is composed of segments of quadratic polynomials (parabolas) and rational quadratics (conics) that join with continuous unit tangent vectors. The boundary is composed of straight line segments and circular arcs.     

有理[2m+1,2m]型分段插值样条

常见的较低次有理带单形状因子分段有理插值样条通过代数运算,可用Bernstein基函数等价表示,这类分段插值样条利用Hermite插值的方法推广到高次有理[2m+1,2m]型,样条的生成曲线满足Cm-连续,并给出了具体的Bern-stein基函数表示方法的表达式,其形式较为简单,最后分别讨论了这类有理插值的逼近阶与约束域及保单调等方面的形状因子的选取情况,并给出了例子分析。     

Möbius变换下四次有理抛物-PH曲线的C2 Hermite插值

目的 曲线插值问题在机器人设计、机械工业、航天工业等诸多现代工业领域都有广泛的应用，而已知端点数据的Hermite插值是计算机辅助几何设计中一种常用的曲线构造方法，本文讨论了一种偶数次有理等距曲线，即四次抛物-PH曲线的C² Hermite插值问题。方法 基于M bius变换引入参数，利用复分析的方法构造了四次有理抛物-PH曲线的C² Hermite插值，给出了具体插值算法及相应的Bézier曲线表示和控制顶点的表达式。结果 通过给出"合理"的端点插值数据，以数值实例表明了该算法的有效性，所得12条插值曲线中，结合最小绝对旋转数和弹性弯曲能量最小化两种准则给出了判定满足插值条件最优曲线的选择方法，并以具体实例说明了与其他插值方法的对比分析结果。结论 本文构造了M bius变换下的四次有理抛物-PH曲线的C² Hermite插值，在保证曲线次数较低的情况下，达到了连续性更高的插值条件，计算更为简单，插值效果明显，较之传统奇数次PH曲线具有更加自然的几何形状，对偶数次PH曲线的相关研究具有一定意义。     

Modifying CAD/CAM surfaces according to displacements prescribed at a finite set of points

This article presents a method for modifying CAD/CAM surfaces automatically in accordance with displacements prescribed at a finite set of points in R³, such as node displacements predicted by finite-element analysis. The method is based on the ‘morphing’ approach introduced by Sederberg and Parry in 1986. The input to the process consists of (a) a CAD/CAM model containing trimmed polynomial B-spline surfaces and (b) a set of points and associated displacement vectors in R³. These points are assumed to be close to, but not necessarily on, the objects of the CAD/CAM model. A rectangular volume, enclosing the CAD/CAM model and the input points in R³, is represented as a volume spline, i.e. a trivariate tensor-product spline. A modified volume spline is computed using (a) a least-squares fit based on the given point displacements, and (b) a smoothing functional. The modified CAD/CAM objects are defined as compositions of the original parametric functions and the modified volume spline (i.e. a morphing). In order to ensure compatibility with standard commercial CAD/CAM systems, the modified surfaces are fitted with appropriate splines using any standard, reasonably shape-preserving, fitting procedure applied in the parameter domains of the original surfaces.     

Functional splines with different degrees of smoothness and their applications

Implicit curves and surfaces are extensively used in interpolation, approximation and blending. [Li J, Hoschek J, Hartmann E. Gⁿ⁻¹-functional splines for interpolation and approximation of curves, surfaces and solids. Computer Aided Geometric Design 1990;7:209-20] presented a functional method for constructing Gⁿ⁻¹ curves and surfaces which are called functional splines. In this paper, functional splines with different degrees of smoothness are presented and applied to some typical problems.     

二次Bézier曲线的双圆弧样条插值二分算法*

在数控加工领域,通常需要用尽量少段数的圆弧样条来对曲线进行拟合。采用二分查找算法,用G1连续的双圆弧样条对二次Bézier曲线进行拟合。该算法在给定误差范围内所需的圆弧段数较少。最后给出了具体的实例说明。